Embedded newform 245.4.x.a.103.37

• Label: 245.4.x.a.103.37
• Level: $245$
• Weight: $4$
• Character: 245.103
• Analytic conductor: $14.455$
• Analytic rank: $0$
• Dimension: $1968$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.4554679514\)
Analytic rank:	\(0\)
Dimension:	\(1968\)
Relative dimension:	\(82\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.37
Character	\(\chi\)	\(=\)	245.103
Dual form			245.4.x.a.157.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.486098 + 0.658639i) q^{2} +(-0.458089 + 0.0866751i) q^{3} +(2.16053 + 7.00426i) q^{4} +(9.36301 - 6.11015i) q^{5} +(0.165589 - 0.343848i) q^{6} +(-9.16435 - 16.0939i) q^{7} +(-11.8448 - 4.14466i) q^{8} +(-24.9313 + 9.78480i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1968 q - 26 q^{2} - 22 q^{3} + 2 q^{5} - 112 q^{6} - 32 q^{7} + 96 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.486098	+	0.658639i	−0.171862	+	0.232864i	−0.882060	−	0.471138i	\(-0.843843\pi\)
							0.710198	+	0.704002i	\(0.248605\pi\)
\(3\)	−0.458089	+	0.0866751i	−0.0881593	+	0.0166806i	−0.229804	−	0.973237i	\(-0.573809\pi\)
							0.141645	+	0.989918i	\(0.454761\pi\)
\(5\)	9.36301	−	6.11015i	0.837453	−	0.546509i
							\(7\)	−9.16435	−	16.0939i	−0.494828	−	0.868991i
\(11\)	−17.1454	+	43.6858i	−0.469958	+	1.19743i								0.477807	+	0.878465i	\(0.341432\pi\)
							−0.947765	+	0.318969i	\(0.896663\pi\)
\(13\)	−0.134434	−	0.0151470i	−0.00286809	−	0.000323156i	0.110530	−	0.993873i	\(-0.464745\pi\)
							−0.113398	+	0.993550i	\(0.536174\pi\)
\(17\)	6.40462	+	0.239644i	0.0913734	+	0.00341895i	0.0830388	−	0.996546i	\(-0.473537\pi\)
							0.00833460	+	0.999965i	\(0.497347\pi\)
\(19\)	−26.6116	+	46.0927i	−0.321322	+	0.556547i	−0.980761	−	0.195212i	\(-0.937461\pi\)
							0.659439	+	0.751758i	\(0.270794\pi\)
\(23\)	3.11860	+	83.3462i	0.0282727	+	0.755604i	0.939267	+	0.343186i	\(0.111506\pi\)
							−0.910995	+	0.412418i	\(0.864684\pi\)
\(29\)	−279.381	+	63.7670i	−1.78896	+	0.408318i	−0.982993	−	0.183642i	\(-0.941211\pi\)
							−0.805967	+	0.591961i	\(0.798354\pi\)
\(31\)	45.5841	−	26.3180i	0.264102	−	0.152479i	−0.362102	−	0.932138i	\(-0.617941\pi\)
							0.626204	+	0.779659i	\(0.284608\pi\)
\(37\)	−319.998	+	169.124i	−1.42182	+	0.751454i	−0.988704	−	0.149883i	\(-0.952110\pi\)
							−0.433118	+	0.901337i	\(0.642586\pi\)
\(41\)	30.4010	+	63.1283i	0.115801	+	0.240463i	0.950811	−	0.309772i	\(-0.100253\pi\)
							−0.835010	+	0.550235i	\(0.814538\pi\)
\(43\)	67.4712	+	192.822i	0.239285	+	0.683838i	0.999419	+	0.0340717i	\(0.0108475\pi\)
							−0.760134	+	0.649766i	\(0.774867\pi\)
\(47\)	23.6631	+	17.4642i	0.0734388	+	0.0542003i	0.629921	−	0.776659i	\(-0.283087\pi\)
							−0.556482	+	0.830860i	\(0.687849\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.x.a.103.37	✓	1968
5.2	odd	4	inner	245.4.x.a.152.37	yes	1968
49.10	odd	42	inner	245.4.x.a.108.37	yes	1968
245.157	even	84	inner	245.4.x.a.157.37	yes	1968

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.x.a.103.37	✓	1968	1.1	even	1	trivial
245.4.x.a.108.37	yes	1968	49.10	odd	42	inner
245.4.x.a.152.37	yes	1968	5.2	odd	4	inner
245.4.x.a.157.37	yes	1968	245.157	even	84	inner